Lamothe SD and "Octave Modularity"

>>>> From a July 11 2001 private communication to Joe Monzo (posted with >>>> Monz's permission):

>>>>Solving for Pierre's "Sonace_Degree" (when including the prime 2) , and >>>>in the general case where the particular unison vectors values would >>>>influence the (summation) determinant's terms associated with primes 3 >>>>and 5 in various different ways, appears to yield some truly puzzling >>>>results. I wondered if you have considered this phenomena, as well. In >>>>a recent tuning-math post responding to Pierre's Message #287:
>>>>
>>>>From: "monz" <joemonz@y...>
>>>>Date: Fri Jun 22, 2001 1:34 am
>>>>Subject: Re: [tuning-math] Re: Sonance degree (DEFINITION)
>>>>
>>>>
>>>> > ----- Original Message -----
>>>> > From: Pierre Lamothe <plamothe@a...>
>>>> > To: <tuning-math@yahoogroups.com>
>>>> > Sent: Thursday, June 21, 2001 11:10 PM
>>>> > Subject: [tuning-math] Re: Sonance degree (DEFINITION)
>>>> >
>>>> >
>>>> >
>>>>PL:
>>>> > Hi Monz,
>>>> >
>>>> > I used abstract language to establish a generalized property, but the
>>>>thing
>>>> > is much more simple than it seems.
>>>> > <etc.>
>>>>
>>>>JM:
>>>> >But this post is indeed very clear! >Thanks so much. I have
>>>> >only glanced quickly at it, but will >study it more fully.
>>>>
>>>>So here's my perplexment. Pierre seems to include the contribution of >>>>the prime number 2 in his main tuning post #18625, Feb 12 2001, where >>>>he states (page 4, bottom)"thus we have to use only 'reduced modular >>>>complexity'" [or,octave-reduced] when he also simultaneously did apply >>>>the contribution of the prime number 2 in his calculations of his >>>>"complexity" and "sonance" (page 4, top).
>>>>
>>>>Yet in his recent tuning-math post #272, Jun 21 2000, "Sonance degree >>>>(DEFINITION)", and post #287, Jun 22 2001, "Re: Sonance degree >>>>(DEFINITION)" he clearly utilizes the number 2 (to the absolute value) >>>>of the exponent E which exist for the various rational (octave-reduced) >>>>possibilities of the "Zarlino" scales rotation in his example. Although >>>>his "Sonance_degree" is normalized by a constant which is proportional >>>>to the term of the (summation) determinant which is associated with the >>>>prime number 2, this value is only a scalar constant.
>>>>
>>>>Here's the rub:
>>>>
>>>>(1) For all distinct rational intervals resulting from such scale >>>>rotation as he describes and cites in his example, IF:
>>>>
>>>>(a) The prime 2 of the (octave-reduced, I presume??) interval has a >>>>zero or positive value, THEN increasing the power of the prime two by >>>>one (equalling transposition of an octave upward for the rational >>>>interval) INCREASES the Sonance_degree, thus REDUCING the intervals >>>>(relative) "con-sonance" (all positive values, and relative to a value >>>>of zero for the 1/1 tonic), ELSE (it must be true that);
>>>>
>>>>(b) The exponent of the prime 2 is of a negative value, AND increasing >>>>the power of the prime two by one (equalling transposition of an octave >>>>upward for the rational interval) *DECREASES* the Sonance_degree, thus >>>>*INCREASES* the intervals (relative) "con-sonance" (all positive >>>>values, and relative to a value of zero for the 1/1 tonic).
>>>>
>>>>What are the implications of such characteristics of the function, I ask?
>>>>
>>>>Is Pierre intending to mean by this functional property that certain >>>>intervals relative to the tonic become MORE "con-sonant" when raised by >>>>an octave, while certain others [depending on their >>>>ORIGINAL(octave-reduced)power of the prime number 2], will become LESS >>>>"con-sonant"???
>>>>
>>>>The ratio (in the case of my calculation for 17 distinct pitches - >>>>which include 1/1, 1/2 and 2/1 - and the 14 intervals centered around >>>>the...1/1 tonic in your version of the 22-note Indian sruti)
>>>>of these two types of intervals as listed above is one-to-one, 8 of >>>>one, 8 of the other! Quite democratic, and non-biased!
>>>>
>>>>Yet, utterly strange. My brain is bent by it, indeed. Could mother >>>>nature be coaxed into yielding more than 50% "con" (rather than "dis") >>>>- "cordance" by simply, it seems to me, sounding its tones within a >>>>(lower) octave (relative to the 1/1), where the exponent of the prime >>>>number 2 is assuredly of a *negative* arithmetic value??? Something >>>>here does not feel right...Or does it? There are certainly MORE >>>>harmonics per unit hertz (CPS) as well as per unit octave in the spectrum,
>>>>when considered relative to the bandwidth equal to the tuning of the >>>>1/1 tonic, and when one sounds the tones of any of the intervals of any >>>>scale progression at a pitch BELOW the reference 1/1 tonic.
>>>>
>>>>But how can such downward octave transposition magically impart values >>>>of the "Sonance_degree" that tend toward "con-sonance" or >>>>"con-cordance" simply by trending downward...Am I missing something >>>>which is *obvious* to others here???
>>>>
>>>>I suppose (thinking out loud) that if they ALL trended downward in >>>>"Sonance_degree" *together*, then all that would matter is each >>>>individual value's magnitude *relative* to some sort of averaging,
>>>>or summation, of ALL of the various values of "Sonance_degree" (SD) >>>>which are present (for the intervals which are being considered).
>>>>
>>>>Ratiometric indicators utilizing the SD of the 1/1 tonic as a reference >>>>point(including logarithms) cannot be taken, since the reference value >>>>(denominator of the arithmetic argument equal to the SD of the 1/1 >>>>tonic) is equal to ZERO...and would be arithmetically undefined.
>>>>
>>>>A linear or root-mean-square average would work nicely, but what would >>>>it mean? If (for, at least in the case of the 14 octave-reduced notes, >>>>relative to the 1/1 tonic from your 22-note Indian sruti diagram), 50% >>>>of the intervals are "con" - "sonant" or "cordant, and 50% are "dis" - >>>>"sonant" or "cordant", then an averaging or summation process would >>>>seem to record only a slow, and seemingly fairly uniform decrease in >>>>the individual various values of SD for each scale interval in the >>>>process, as all of those scale intervals (which are relative to some >>>>reference tonic) would in such a case be subject to...*equal* (it has >>>>been presumed)... decreases in pitch on an octave basis. And, it would >>>>seem, thus would follow an uneventful monotonic decreasing trend of >>>>each of the individual values of a ("normalized" by average/summation) >>>>"Sonance_degree" for each of those individual value considered
>>>>(where some averaging/summation process provides a reference value >>>>which is responsive to a summation of the variations in the individual >>>>values of SD for each scale interval which is undergoing a shift >>>>downward (or upward) by 2^(integer) power.
>>>>
>>>>If Pierre meant for SD to be only valid for "octave-invariant" (reduced >>>>or increased) ratios between 1/1 and 2/1 (or possibly between 1/2 and >>>>2/1) then WHY does he even...include it (the prime number 2 and it's >>>>exponents)...in his example calculations? It seems there must be reason.
>>>>
>>>>If, however, one (or a group of any number which is less than all of >>>>the other tones, excepting the 1/1 tonic) of the intervals is reduced >>>>in pitch by integer powers of the prime number 2 ONLY, and the others >>>>are *NOT* (as in playing the musical 5th, as well as any group of other >>>>tones, at a pitch below the tonic by some number of sub-octaves, etc.), >>>>then we have in such a (normalized, or relative)"Sonance_degree" >>>>(NSD?), as detailed above, an exotic, and *octave-variant*, numerical >>>>quantity which would seem to (in some way) represent the "con" or "dis" >>>>- "sonance" or "cordance" of multiple simultaneously sounded tones, >>>>relative to the 1/1 tonic.
>>>>
>>>>At that point,...why not apply any chord within the scale (with any of >>>>the intervals of the chord played in any octave), and...just...watch >>>>the "sonance"/"cordance" meter!?!?
>>>>
>>>>I ask you, my friend, is this *too* good to be *true*, or, has mother >>>>nature actually been so tamed by Pierre Lamothe's algorithm?
>>>>
>>>>Or has Gill simply stumbled across only what seems intuitively obvious >>>>to certain other amazing minds (such as Lamothe's)?
>>>
>>>PS- Or, has Gill, in his nieve passion, blithely ignored the total of >>>N^2-N other combinations of intervals present (where N equals the number >>>of degrees of the scale)? Nevertheless, all of the interactions present >>>between the simultaneously sounded tones (and complex tones with energy >>>at all multiples of each fundamental included, at that!) could be >>>calculated. The ultimate mystery is not whether such numerical values >>>could be rigorously calculated by a computer, but, instead, it is the >>>question of whether human auditory perception, being a (rather) highly >>>non-linear (and, in some respects, time-variant coefficient) process >>>actually treats such complex spectral input (as our "wonder-machine" >>>could) as a linear *superposition* of effects, in like manner to our >>>machine, the ear being a logarithmic device in amplitude (log/log slope >>>of apporximarely 2/3), which creates additional sum and difference >>>products at sound levels experienced in performance/listening [this is >>>essentially an "inter-modulation distortion" of harmonic and >>>non-harmonic distortion products (sum and difference frequencies) of the >>>multiplicative co-mixing of each of the individual spectral component >>>with *all* of the other spectral components].
>>>
>>>I read that "loudness was found to increase perceptual judgements of >>>dissonance" (Plomp, Levitt, 1969, and Kameoka and Kuriyagawa a few years >>>later), the latter also reporting that the "perceived dissonance of two >>>single partial tones was greater when the low frequency tone was of >>>greater amplitude than the high frequency tone", which consistent with >>>inter-modulation distortion as it typically occurs, due to the sound >>>pressure level (SPL) increasing by an exponent of between 1.5 and 2.0 >>>for each decrease in frequency of one octave (an exponent of 1.5 being a >>>"pink", or equal energy-per-octave, noise spectrum).
>>>
>>>I read that in 1939, Sandig found that "dichotic" (stereo) "experiments >>>of consonance perception showed that dissonant tones are perceived as >>>nuetral when split into two sets of harmonics presented to seperate >>>ears, so who's going to ever be able to really state what's going on?
>>>
>>>The only thing that seems clear, and this seems problematic for a number >>>of lines of thought which (for simplicitys sake) do not factor in the >>>partials of each individual note simultaneously sounded (as such notes >>>are generated by real-world tone generators, excepting flutes and >>>synthesizers),
>>>is that combinations of two sinusoidal wave forms of various frequency >>>ratios have been reported (by Plomp and Levelt, 1965) to, "not play any >>>significant role when presenting pairs of pure tones", and, in other >>>experiment(s) in the same year, reporting that pairs of sinusoidal tones >>>at "critical band" frequency ratios, "the two sine waves interact to >>>produce dissonant sensations". It would seem to me, based upon the >>>limited set of studies relied upon herein, that, with exception to such >>>"critical band" frequency ratios between two sinusoidal tones, the >>>"sonance" or "cordance" relies entirely on the prescence/absence of >>>energy which is attributible to the partials of those fundamentals >>>existing together with the spectral energy of those two fundamentals (in >>>the two-tone case, at least), and where the octave in which the >>>(complex, real-world) tone is sounded (relative to the 1/1 tonic) cannot >>>ultimately be ignored in analysis.

>>>>What are your thoughts on this?, J Gill

>>>> PS - I am very interested in anyone's (included Mr Lamothe's, if he >>>> is able to find the time) feedback on this.

In post #496, #497, #499 ... J Gill wrote

<<
So here's my perplexment. Pierre seems to include the
contribution of the prime number 2 ...

-- SNIP --

I am very interested in anyone's (included Mr Lamothe's,
if he is able to find the time) feedback on this.
>>

It's like a BANG BANG BANG ... at my door. I imagine I don't have the
choice to make an exception and answer. However, I would avoid to quote
more over all.

:-)

About the post #18625 on the Tuning List, the 'reduced modular complexity'
was a temporary notion having utility only to show the effect of the
representation in Z-module using octave modularity. That modularity is
expressed by a rotation, as I wrote to Paul Erlich in post #18656

<< As you may know, my definition of the sonance (or log
complexity)

C = C(X) = |x|*log 2 + |y|*log p + |z|*log q

corresponds to the Tenney's Harmonic Distance. What I
wanted to show is the effect of the representation of
this sonance by the distance

MC = MC(X) = |x| + |y| + |z|

in <2,p,q> as Z-module and the distance

RMC = RMC(X) = |y| + |z|

in <p,q> where octave modularity is used.

I underline that it is not question of complexity metric
but topological invariance using MC and RMC to represent
sonance (or log complexity). The first reduction corresponds
to a variation in axis elongation ratios and the second to
a counterclockwise rotation. The invariance concerns the
spatial location of the intervals between them and the
convexity property is not changed by these two
transformations. >>

While MC and RMC had only a temporary use, the _Sonance degree_ is rather a
significant concept having sense in all congruent system (using unison
vectors) having a Degree function (a surjective morphism on Z) preserving
both the 'width order' and the 'sonance order' of the intervals. Since I
did'nt precise that in the definition, I will give in APPENDICE the
condition to preserve the sonance order.

Comparing here the _Sonance degree_ (SD) using the operator D = [k2 kp kq]
with the precedent definitions of sonance (S) and temporary MC and RMC

SD * k2 = |x| * k2 + |y| * kp + |z| * kq
S * log(2) = |x| * log(2) + |y| * log(p) + |z| * log(q)
MC = |x| + |y| + |z|
RMC = |y| + |z|

we can see that only the _Sonance degree_ is an approximation of the
_Sonance_ and as long as log(2):log(p):log(q) is well approximated by
k2:kp:kq.

I would E M P H A S I S here these remarks :

(1) Like the Sonance (S), the _Sonance degree_ (SD) concerns
first the intervals and it is applied to octave classes
only in a derived manner. The value SD for an interval
class modulo 2 is determined (by convention) as the value
SD for the interval in the first octave of this class.

(2) The variation of SD with octave changing is not proper to
SD. It is strictly the same as the variation of S. So if
that variability is a source of perplexity, this has got
nothing to do with the specificity of SD as approximant
of S but with the Sonance (Tenny's distance) as such.

-----

The variation of the sonance with the octave changing is independant of the
approximation of S by SD, and the sonance of an irreducible ratio n/d
between any rational pair (kn:kd) is both independant of any musical
context and any acoustical spectrum. It is simply

(log n + log d) / log 2

which is the log in base 2 of the complexity n*d.

-----

The following table shows for the classes of 9/8, 6/5, 5/4 and 4/3 the
distinct values in 7 octaves for ratio, complexity, and the variation of
the sonance S (or sonance degree SD since the variation is the same)
compared to

a = S(9/8), b = S(6/5), c = S(5/4) and d = S(4/3)

in the first octave (i.e. the values used for the classes).

-3 -2 -1 0 1 2 3

9/64 9/32 9/16 9/8 9/4 9/2 ( 9/1 )
3/20 3/10 ( 3/5 ) 6/5 12/5 24/5 48/5
5/32 5/16 5/8 5/4 5/2 ( 5/1 ) 10/1
1/6 ( 1/3 ) 2/3 4/3 8/3 16/3 32/2

596 288 144 72 36 18 ( 9 )
60 30 ( 15 ) 30 60 120 240
160 80 40 20 10 ( 5 ) 10
6 ( 3 ) 6 12 24 48 64

a+3 a+2 a+1 a a-1 a-2 ( a-3 )
b+1 b ( b-1 ) b b+1 b+2 b+3
c+3 c+2 c+1 c c-1 ( c-2 ) c-1
d-1 ( d-2 ) d-1 d d+1 d+2 d+3

The minimal sonance inside a class occurs for the "pivot" of the class
which are here ( 9/1 ), ( 3/5 ), ( 5/1 ) and ( 1/3 ). Increasing or
decreasing by N octaves relatively to the pivot results to add N to the
sonance value of the pivot.

It's there a microtonal property (i.e. independant of the organisation of
the intervals). Besides, for a strutural understanding of musical modes it
is sufficient to use the conventional sonance value attributed to the
interval in the first octave which represents the class. The microtonal
property is not neglected here since it is well reflected in the
classification +/- major or minor of the modes (inside the structure).

An interval like 9/8 is said "major" for its pivot 9/1 is higher while an
interval like 8/5 is said "minor" for its pivot 1/5 is lower. So a major
interval is more consonant an octave above while a minor interval is more
consonant an octave under. A mode like 1 9/8 5/4 4/3 3/2 5/3 15/8 is major
relatively to its dual 1 16/15 6/5 4/3 3/2 16/9 for "major" intervals
predominate in the first mode while (forcely) "minor" in its dual.

-----

The conventional use of the intervals in the first ascending octave to
represent the class modulo 2 introduce an asymmetry for the complexity
between an interval like 5/4 and its reverse 8/5. A symmetric complexity
would result if pivots were used or if the conventional octave of reference
was centered

1/sqrt(2) < octave < sqrt(2)

When I need a symmetric sonance for the classes in particular context, I
use simply the sum

SS(5/4) = SS(8/5) = S(5/4) + S(8/5)

which would be the same with the first descending octave representing the
class

SS(4/5) = SS(5/8) = S(4/5) + S(5/8).

-----

Any acoustical consideration here seems to me at hundred miles of the
question. There exist, for sure, acoustical conditions permitting to
perceive an interval and contextual conditions permitting to perceive it as
a category. Inside the limits of these good conditions, we need to change
the semiotic level to understand relational properties like sonance
(microtonal) and sonance degree (macrotonal).

Since this post is an exception, I would add, without translation, what I
wrote to Yves Hellegouarch, almost a year ago :

<<
Aussi justes soient-elles, les considérations acoustiques,
en regard de la question des gammes, m'apparaissent
piégeantes. Elles introduisent subrepticement une erreur
de perspective. La place privilégiée qu'occupe la notion
d'harmonique dans le son musical laisse croire à une semblable
primauté au niveau des structures tonales. Et on confond
généralement trois aspects de l'harmonicité, selon qu'elle a
trait à la paramétrisation du son musical, à l'explicitation
de la sonance des accords, ou à la génération des modes
musicaux. Or l'harmonique n'occupe une place privilégiée
que dans le son musical. Mais alors pour cause.

Pour qu'apparaisse l'intervalle de hauteur, à titre d'atome
d'intelligibilité dans des processus d'évaluation de
similitude formelle, il faut que les sons soient musicaux,
autrement dit, qu'ils possèdent une périodicité principale
marquée. Et c'est en raison seulement de cette périodicité
qu'un tel son est analysable en terme d'harmoniques (et non
de sous-harmoniques) de la fréquence principale.

Au plan de la sonance, les théories acoustiques de la
dissonance, de l'alignement des harmoniques d'Helmholtz aux
bandes critiques de Zwicker, privilégient aussi l'harmonique.
Mais j'estime qu'elles sont foncièrement erronées, au sens
où elles peuvent espérer rendre compte de la rugosité mais
nullement de la dissonance. Enfin, au niveau macrotonal, que
j'estime avoir adéquatement théorisé, autour des problématiques
de la liaison de la cohérence à la consonance puis de
l'émergence de la simplicité, rien ne peut plus permettre de
privilégier le générateur harmonique en regard de son dual.
Mieux, sans la dualité, la musique ne serait probablement que
rythmique et peinture sonore.

>>

-----

About the << mother nature >> I would quote also a text on my website

<<
Il faut savoir apprécier les contorsions mentales auxquelles
on devait se livrer pour justifier la triade mineure à
partir des idées de Rameau, puis cette assurance d’Hindemith
qui se voyait écraser le dualisme de Riemann d’un simple
diktat à croire en béton : « les sous-harmoniques n’existent
pas dans la nature ». On commence à peine à comprendre que
la linéarité (autrement dit le fait qu’on dispose de
composantes qui n’interfèrent pas entre elles) n’est pas
imposée par la nature — bien qu’elle ait à voir avec le
phénomène physique de la résonance — mais par l’esprit.

C’est l’exigence de périodicité qui détermine et non une
quelconque tendance de la nature à créer des sons harmonieux,
dont on se demande bien d’ailleurs, mis-à-part l’exemple des
oiseaux, où ils seraient présents. Depuis la découverte du
chaos déterministe on prend conscience, surtout, que la nature
ne se contente pas des scénarios qui paraissent les plus
simples, et que cette omniprésence de la linéarité dans les
sciences physiques ne reflète en rien une prééminence
naturelle séparée de notre esprit.
>>

Pierre Lamothe

APPENDICE

I did'nt precise in my _Sonance degree_ definition the conditions about the
unison vectors giving a true coherent system. I added here that the resulting
Degree function X --> D(X) has to be a surjective morphism on Z preserving
both the 'width order' and the 'sonance order' of the intervals.

It's not the good place to develop my theory of the coherent unison
vectors. But since SD is in question I would like to precise that the
condition which preserves the 'sonance order' is simply that the canonical
basis have integer values. I wrote in post #272 about the definition

<<
in the basis

B = <B0 B1 .. BN>

where B0 = 2 (octaviant system) and the other independant
components are normally simple primes (primal basis).
>>

Normally, in simple systems, the canonical basis is a primal basis. But
there exist Z-modules in which links exist between primes. The two types of
such links are caracterized by these examples

<2 3*5 7> Z^3 is a submodule of <2 3 5 7> Z^4

<2 3 7/11> Z^3 is a submodule of <2 3 7 11> Z^4

There is no problem in the first case whose canonical basis <2 15 7> has
integer values while a D function associated with a periodicity block in
the second case don't preserve the 'sonance order' of intervals.

Scales of Margo Schulter using third 14/11 rather than 5/4 are concerned
with that but it is sufficient to show here the pentatonic system generated by

<7 11 33 99> == <1 3 9 7/11>

which is an approximant or the 5-limit chinese system <1 3 5 9> like the
Ling Lun <1 3 9 81> is a such approximant.

With a canonical basis such <2 3 7/11> a Sonance degree (SD) has no sense
since the sonance S of (x,y,z) is

|x|log(2) + |y|log(3) + |z|log(7) + |z|log(11)

and not

|x|log(2) + |y|log(3) + |z|log(7/11)

which is equal to

|x|log(2) + |y|log(3) + |z|log(7) - |z|log(11)

The systems where we find scales near to Margo Schulter's ones don't
preserve the 'sonance order', but it's not incoherent with the Gothic style
of music using fifth and fourth as "stable sonorities". It would be
different trying to consider the approximant chord 22:28:33 as a "stable
sonority" like its chord image 4:5:6 in <1 3 5 9>.